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1-4 SOLVING EQUATIONS
Properties of Equality
Reflexive Property of Equality Symmetric Property of Equality Transitive Property of Equality Substitution
PROPERTIES OF EQUALITY
Reflexive Property
• a + b = a + b
The same expression is written on both sides of the equal sign.
PROPERTIES OF EQUALITY
• If a = b then b = a
• If 4 + 5 = 9 then 9 = 4 + 5
Symmetric Property
PROPERTIES OF EQUALITY
Transitive Property
• If a = b and b = c then a = c
• If 3(3) = 9 and 9 = 4 +5, then 3(3) = 4 + 5
ADDITIONAL PROPERTIES OF REAL NUMBERS
Substitution Property
• If a = b, then a can be replaced by b.
• a(3 + 2) = a(5)
NAME THE PROPERTY
• 5(4 + 6) = 20 + 30• 5(4 + 6) = 5(10)• 5(4 + 6) = 5(4 + 6)• If 5(4 + 6) = 5(10) then
5(10) = 5(4 + 6)• 5(4 + 6) = 5(6 + 4)• If 5(10) = 5(4 + 6) and
5(4 + 6) = 20 + 30 then 5(10) = 20 + 30
• Distributive• Substitution• Reflexive• Symmetric
• Commutative• Transitive
MORE PROPERTIES OF EQUALITYIF THE OPERATION DONE TO ONE SIDE IS ALSO DONE TO THE OTHER THEN
THE VALUE OF THE EQUATION DOES NOT CHANGE
Addition: If a=b, then a + c = b + c
Subtraction: If a=b, then a – c = b – c
Multiplication: If a=b, then a ∙ c = b ∙ c
Division: If a = b, then a / c = b / c (c≠0)
If x = 12, then x + 3 = 12
+ 3 If x = 12,
then x – 3 = 12 – 3
If x = 12, then x ∙ 3 = 12 ∙
3 If x = 12,
then x / 3 = 12 / 3
Definition Examples
SOLVE THE EQUATION
Solving an equation that contains a variable means finding all the possible values that make the equation true. The first step is to isolate the variable to one side of the equation by using inverse operations.
Inverse operations undo operations. Addition, subtraction are inverse operations
as are multiplication, and division .
Solve . Check your solution.
Original equation
Add 5.48 to each side.
Simplify.
Check: Original equation
Answer: The solution is 5.5.
Simplify.
Substitute 5.5 for s.
Example 1
YOUR TURN
What is the solution of 12b=18?
12b / 12 = 18 / 12 Divide each side by 12
b = 3 / 2 Simplify
Solve
Original equation
Distributive and Substitution Properties
Commutative, Distributive, and Substitution Properties
Addition and Substitution Properties
Division and Substitution Properties
Answer: The solution is –19.
YOUR TURN
What is the solution of -27 + 6y = 3(y – 3)?
-27 + 6y = 3(y – 3) -27 + 6y = 3y – 9 Distributive
Property 6y = 3y + 18 Add 27 to each side. 3y = 18 Subtract 3y from each
side y = 6 Divide each side by
3.
YOUR TURN
What is the solution of 3( 2x – 1) – 2(3x + 4)=11x?
3( 2x – 1) – 2(3x + 4)=11x 6x – 3 – 6x – 8 = 11x Distributive Property – 11 = 11x Combine Like Terms – 1 = x Divide each side by –
11 x = – 1 Symmetric Property
Suppose the flower carpet from Problem 3 had a perimeter of 320 meters. What would the dimensions of the flower carpet be?
YOUR TURN
PROBLEM 4 PAGE 29
PROBLEM #5 PAGE 29
ANOTHER LITERAL EQUATION
Distance = Rate x Time or d = r ∙ t
Solve for r
Solve for t